We give a characterization of Possonian domains in Rⁿ, i.e., those domains for which every bounded harmonic function is the harmonic extension of some function in L^∞ of harmonic measure. We deduce several properties of such domains, including some results of Mountford and Port. In two dimensions we give an additional characterization in terms of the logarithmic capacity of the boundary. We also give a necessary and sufficient condition for the harmonic measures on two disjoint planar domains to be mutually singular.

Let ℒ(H) denote the space of operators on a Hilbert space H. We show that the extreme points of the unit ball of the space of continuous functions C(K, ℒ(H)) (K-compact Hausdorff) are precisely the functions with extremal values. We show also that these extreme points are (a) strongly exposed if and only if dim H<∞ and card K<∞, (b) exposed if and only if H is separable and K carries a strictly positive measure.

An open subset D of R^d, d≧2, is called Poissonian iff every bounded harmonic function on the set is a Poisson integral of a bounded function on its boundary. We show that the intersection of two Poissonian open sets is itself Poissonian and give a sufficient condition for the union of two Poissonian open sets to be Poissonian. Some necessary and sufficient conditions for an open set to be Poissonian are also given. In particular, we give a necessary and sufficient condition for a Greenian D to be Poissonian in terms of its Martin boundary.

In this thesis, we consider some aspects of noncommutative classical invariant theory, i.e., noncommutative invariants of the classical group SL(2, k). We develop a symbolic method for invariants and covariants, and we use the method to compute some invariant algebras. The subspace Ĩ ${}_{d}^{m}$ of the noncommutative invariant algebra Ĩ_d consisting of homogeneous elements of degree m has the structure of a module over the symmetric group S_m. We find the explicit decomposition into irreducible modules. As a consequence, we obtain the Hilbert series of the commutative classical invariant algebras. The Cayley—Sylvester theorem and the Hermite reciprocity law are studied in some detail. We consider a new power series H(Ĩ_d, t) whose coefficients are the number of irreducible S_m-modules in the decomposition of Ĩ ${}_{d}^{m}$ , and show that it is rational. Finally, we develop some analogues of all this for covariants.